 So, today's topic is directly abstract simpial complex, abstract simpial complex. Last time we introduced from affine geometry, let me before going to abstract simpial complexes, one more or one or two more concepts from affine geometry, let us cover that and then come to affine, this abstract simpial complexes. So, here is, this theorem was done, here is a definition, convex combination of these points, n points in solidarity, it is an affine combination lambda xi, summation lambda equal to 1, but one more condition, each lambda i must be positive non-negative also. Automatically, each lambda i will be less than equal to 1 also, because sum total is 1. So, such thing is called convex combination, just like 0 less than equal to t less than equal to 1, I have been taking t times p plus 1 minus t times q, okay, there it is convex combination of only two elements, but here you could take any finite set, then you can talk about convex combination. The convex hull is, you know, set of all convex combinations, take set any edge, then you can take convex hull of that, it is called a convex, con of m, that is a notation. It is automatically a subset of affm, affine m, here all linear combinations are taken without lambda i greater than equal to 0, this condition is not there, okay. So, a subset a of r is said to be convex, if convex hull of a itself is equal to a, it is similar to what in topology, something is closed, its closure is equal to a, it is like that. What is the meaning of that? If you take a convex combination of finite to many points of way, it is again inside a, it is similar to affm affine subspaces, okay, only convex subsets here. So, line segment is a convex set, not a line. Similarly, the inside of a triangle full, not just the boundary triangle, inside of a triangle is a convex set, tetrahedral. These are our standard convex subsets, okay. There are many, even a square is a convex set, right. So, you will see that we are concentrating on particular kind of convex sets soon, okay. So, this is what is called a geometric n-simplex, geometric n-simplex. So, by the very depth name, you may anticipate that there is going to be something n-simplex also without that tag geometric. So, what is geometric n-simplex? It means convex of any n plus 1, affinely independent points v1, v2, vn plus 1 in all elements of v i are called vortices of the simplex. The number n is called the dimension of this a. So, a is a convex of v1, v2, vn plus 1, v1, v2, vn plus 1 must be affinely independent. Then we call it as geometric n-simplex. A zero-simplex is nothing but a single point. To take one single point, what is its affine combination? Lambda times x, where summation lambda must be 1 now. So, it is just lambda is 1. So, it is just x itself. So, its convex combination is also one point, okay. And one simplex is a line segment. A two simplex is a triangle, a three simplex etc. These are names. Afterwards, we do not know what to do. The four simplex, I do not know what name I should give. So, we just go n simplex, okay. Note that every element of a, the unique combination of these vertices, uniquely, unique convex combination, okay. So, uniqueness follows because of affine independence of v1, v2, vn. If T i times v i equal to summation s i times v i in two different ways, then what you will get is s i minus T i times v i will be 0, which at least one of the s i minus T i will not be equal to 0. Sum total of s i minus T i is equal to sum total of s i plus minus sum total of v i which is 1. So, that will give you that v i's are not affinely independent. Therefore, uniqueness follows. It is exactly similar to whenever you have a linearly independent set, then every element in the linear span of these sets is a unique linear combination of these, the n elements there. That is called a basis there, right. Here, these v i's are called vertices. There is one very special point, you know, by symmetry. What is it? It is summation v i divided by n plus 1. There are n plus 1 points here. So, I am dividing by n plus 1 after taking a summation. So, this is a convex combination and this point is called the bary center of a, okay. You may call it centroid also, okay. There are lots of centers, not in a plane geometry, right, ortho center, incenter and so on, various centers. We have just bary center will do. This bary center is called, some people call it centroid also, no problem. Here is an easy theorem. Let a and b be any two geometric simplices, okay. There exists an affine isomorphism from a to b such that f a equal to b and a and b are isomorphic in that sense, affine isomorphism. You find only dimension of a equal to dimension of b. So, just the dimension will tell you that they are affinely isomorphic, okay. So, this is the starting point, maybe you can say that, you know, the topology here is reduced to permanent oryx, finite geometry. So, this is the starting point of that. So, you will see more and more of this thing. Let us see how to prove. Suppose dimensions are the same. Choose some labeling, that means number of vertices are the same, right. a1, a2, ak and v1 will be k. Dimension here, it will be actually equal to k is dimension of a plus 1, one more. So, that remember, that is what we have seen the dimension by definition. Take f of a i equal to b i, it is a one-one correspondence of vertices. Choose any one-one correspondence, okay. Extends it linearly over, extend it linearly means what now? Affine linearly. Lambda i, a i summation will go to lambda i, b i summation. That is all. Just like in linear algebra, okay. Once you have defined it over basis elements, it is uniquely defined in linear algebra exactly here also. It is uniquely defined because each lambda i, v, a i has unique expression, okay. So, that will give you an isomorphism. So, conversely, if a to b is an isomorphism, then a i's are affinely independent. f of a i should be affinely independent. Similarly, f inverse of b i should be affinely independent. Therefore, the dimension of a that will be k and l, a must be the same, okay. So, this is all missing last time we came to it. So, I have done it. Now, we will directly take up the study of abstract simplicial complexes which does not look like any topology at all you will see now, okay. So, you have to hold your hearts for a while to see the topology behind them, all right. So, I will directly start with the definition of abstract simplicial complex. It will consist of a set denoted by v and called as vertex set and another set denoted by s called what is finite collection of subsets simply saves, okay. I will write down this one later on. s is a collection of finite subsets of v, okay. It satisfies only two of the conditions namely for every v inside v the singleton is inside s remember elements of s are subsets of v, okay. So, I should not write v inside s but I should write singleton v inside s whenever f is a subset belonging to s all subsets of f will be also inside s in other words s is closed under taking subsets, okay. So, these are the only two conditions that will make a simplicial complex. So, let us go on do some more definitions and properties. Elements of v are called vertices of k those of s are called simplicies of k. If f is inside s number of points namely cardinality of f say equal to q plus 1 then f is called a q simplex, okay. And you can also say dimension of f is q. So, this is the definition of dimension get the number of points minus 1. Thus the vertices of k are all zero simplicies of k. You take a vertex put it inside a bracket to make it a singleton it will become a zero simplex. So, soon we will not you know we will have this liberty of not writing the bracket that is all like we write sometimes v minus zero where v minus singleton zero. So, some set theory we take such that kind of liberty you know. So, only that liberty is there that is all. Thus vertices of k are zero simplicies, okay. So, as a vertex they are elements of v, but as elements of s it is a singleton v's which are inside a state sort that is the difference between logically what it but you can say that the vertices also zero faces, okay. So, if f prime is contained inside f and f itself is in s then we say that f prime is a phase of s. A phase here it may be equality also, okay. So, for example, if you take one vertex here a singleton that singleton v will be a phase. So, all subsets are called phases. Often a simplex is described by enumerating its zero spaces. Enumerating zero phases means what? Just writing down the set that's all. What are the elements of that set? Just write down automatically all the subsets are there is there you have to think about it that's all. This is just like writing when when you have topology you just write down sub basic open sets only then the topology is understood. Observe that we have followed allowed empty subset also as simplex. I never said f is not empty. Okay empty set is also allowed. What is the dimension of empty set? There are no elements there. So, cardinality is zero then I have to subtract one. So, the dimension is by definition minus one, okay. Finally, whenever the vertex set itself is finite then automatically s will be also finite because it's only finite subsets of a finite set, right. Therefore, that will be also finite. So, k itself is called a finite simplex. The vertex set must be finite. So, these are just some names I will keep reminding you again and again. So, soon you will get used to it. Now, I can assign the dimension to the whole of k. The k is an inferior complex. So, what is this dimension? It is the supremum of all ns where n is for some simplex of dimension of some simplex in k. Suppose there is a simplex of dimension n, then dimension of k will be at least n. So, I have to take the supremum. This supremum could be infinite. If there is a simplex of dimension n for every n then dimension of k will be finite. If it is finite means, okay, say k, what is that dimension? There is at least one simplex of that dimension and all other simplex is of dimension less than or equal to that n. It is a supremum. It is finite. The finiteness of k, dimension is k, does not mean that k is finite. But if k itself is finite, then the dimension has to be automatically finite, okay. If the dimension of k is finite, then there are simplex of dimension n and all simplex of dimension are these are maximal. Maximal means what? You cannot have another simplex containing that. If one simplex containing that, the dimension would have increased, right. So, these simplex will be maximal, which means all other simplex are subsets of simplex of dimension here, okay. Not all maximal simplex need to be of the same dimension. Remember, I can have a singleton zero, singleton say one here and then a simplex somewhere, okay, both will be maximal. Dimension will be one here, but this singleton zero standing away, that will have dimension zero, okay. So, whenever everything is contained in n simplex, such a simplex is called pure. These things are very, very important in algebra and combinatorial mathematics. But for us, this will not play much role, okay. So, you can soon forget about this one. Perhaps I am never going to use this word pure. Simplicial map, see. You see, now we have introduced the definition of simplicial complex, okay. Now, we want to introduce the notion of what is the map between them. Like topological space is this continuous map. If they are groups, there is homomorphism and so on. So, we should have relations also, what kind of functions are allowed here. So, take a simplicial map from k1 to k1 means what, okay. So, k1 itself has, you know, structure of k1 means what? It has a v1, a vertex, an s1, a set of stabilizes, right. So, the function phi corresponds to actually a function from, saturated function from vertex at v1 to vertex at v2, okay. With an additional hypothesis shown that, condition that each simplex f inside k1, phi of f makes sense because phi is a function of v1. So, phi of f makes sense. This is a subset of v2. It must be inside, k2 means what? It is simplex in k2. It must be inside s2. f belongs to s1 means phi of belongs to s2, okay. So, that is the condition. Image of every simplex must be a simplex. It need not be of the same dimension. Number of elements in f may be say 5. Number of elements in phi f may be 4. It cannot be more than 4. We know, we, more than 5, we know, okay. It may be 4. It may be 3. It may be 2 also. But phi being function, it will not be empty. That is all. If f itself is empty, phi empty will be empty. That is no problem, okay. So, usual saturated composition of functions, t from v1 to v2, another one say psi from v2 to v3, then psi compose the phi makes sense. Automatically, it will be simplexial because psi of phi f, this is as a simplex, must be a simplex inside s3, okay. So, composition of simplexial maps is a simplexial map. Identity map from k2k namely vertex to vertex is a simplexial map. No problem, okay. What is the meaning of simplexial isomorphism? Simplexial isomorphism is first of all a bijection on the vertex sets. That means there is a inverse vertex set. The inverse must be also a simplexial map. Inverse must be a simplexial map, okay. A bijection of vertex sets, okay, need not be an isomorphism even if it is a simplexial map. The inverse may not be simplexial. So, this is strangely typical of topology. A continuous bijection may not be a homeomorphism. The inverse may not be continuous. Whereas in linear algebra, we have a linear bijection. Inverse is automatically linear. In group theory, if you have a linear, if you have group homomorphism, it is a bijection. The inverse is automatically a group homomorphism, okay. So, though this is like combinatorics, this is like algebra, it is more close to topology in this nature. Inverse has to be, you know, I have to demand that inverse is also a simplexial map for an isomorphism. Some more terminologies. What is the meaning of a subcomplex? Subcomplex k prime, say v prime, s prime of a simplexial complex k, okay. We will just denote it by k prime sub of k. So, what is the meaning of this? The set vertex set v prime must be a subset of v and s prime must be a subset of s, okay. k prime itself must be a simplexial complex on its own, first of all. And these inclusions must be valid. Then only you call it a subcomplex. What is the meaning of this? The vertex set must be a finite, must be a subset of the original set v. Whenever you have simplex inside k prime, it must be simplex inside chaos. That is the meaning of that. All the vertices of v prime, okay, are inside v. You may, for example, take all the vertices of v equal to v prime. But suppose some of the vertices in s are missing here. That is the way of taking a subcomplex. It will be a subcomplex. Some of them are missing, but whatever you have taken, they are inside s. You do not have taken, you do not take new simplices, okay. So, that will be a subcomplex. Note that in this situation, the inclusion map k prime to k is a simplexial map because every simplex in k prime is a simplex here and the map is inclusion. Phi is inclusion map here, okay. So, I remember, I recall k prime itself is a simplexial complex, okay. By subcomplex, it is a simplexial complex. You know, we mean a simplexial complex k prime. What is the meaning of this? 1 and 2 must be satisfied. That whenever actually number 2 is most important. Whenever say f is in k prime and f prime is subset of f, all that f prime must be also in k prime. That must be automatically here. So, you cannot say that you keep one subset here and miss a subset of that set. That should not happen, alright. So, let us run through a few of examples now. I may not be able to do all of them. Take any set, okay. Then take the collection of all finite subsets of that. s is all finite subsets. So, that is a simplexial complex. So, this corresponds to like we take, you know, in topology what do we do? We take a set and then take all subsets, right. That is a discrete, right. We are taking all subsets here which are finite of course because we do not, we are not allowed to take infinite subsets. All finite subsets to take. That is a simplexial complex. This simplexial complex contains, as subcomplex says, all simplexial complexes vertex set is a subset of v. You cannot have anything better than that. That is why I told you that this is like a discrete space, okay. In logic it is discrete but you will see that topology later on it is the opposite. Isomorphism type of this simplexial complex depends only on the cardinality of v, okay. I just take v and I just take w. If the cardinalities are the same, then taking all subsets here, taking all subsets here, there will be a corresponding isomorphism. All that I have to do is write down a bijection of the vertices. That is all. Take any bijection of vertices. It will be a simplexial isomorphism. Sir, the dimension would be one plus cardinality of v, right? Dimension of s of this k. Yes. Dimension of this k could be finite. Depends upon what is v. Suppose v is infinite set. Then I can go on taking finite subsets, okay. If v is a finite set, suppose there are only n elements, then the dimension will be n minus 1, okay. If it is infinite, the dimension of this simplexial complex will be infinite. Yes sir. Each simplex is finite, finite dimension but dimension of k is infinite. Let us go to a second example. Consider a special case where v itself is finite. That is what we are discussing, right? With n plus 1 elements. This is a very important one for us. Now, what are you taking? You are taking n plus 1 element set. Then you are taking all subsets, empty set, then all the singletons, then all the double tons, all the triple tons and so on. You are taking, okay? Yes or no? Yes. This is, this is now characteristic of our, what we are calling that standard n simplex. What we do? Take the basic unit vectors e1, e2, en plus 1 inside rn plus 1, okay? We denote this corresponding simplexial complex by the simple delta n. Now, I am giving you another specific case. Instead of arbitrary set v, I am taking v to be even e2en plus 1, okay? Then take all subsets of even e2en plus 1. That simplexial complex we are denoting by delta n without the bars. Converse combination, remember converse combination of these set was denoted by more delta n. So soon we will understand what is the difference between these two symbols, okay? Right? So this overuse of the same symbol is deliberate because they are very closely related. So you have to hold your horses till, how to, what is the difference between them? What I want to say is, again I repeat, take any simple, any simplexial complex with this property, namely n plus 1 point and then all subsets. They are all isomorphic to this delta n because our previous thing here, this remark is applicable here also. Take any bijection of the vertex state, that will give you an isomorphism. So up to isomorphism and n simplex, okay? The delta n is unique with a single one. Any simplexial complex of finite dimension can be described by declaring all maximal simplices in it, okay? If it is infinite dimension, there is no maximal. So that is it, you can do it only for finite dimension case. Look at what are all the maximal simplexes. I will give you a list of them. Rest of them, you can know what are the other simplices, they are subsets of this, that is all. In fact, all subsets of whatever I have given you, they must be there. So this is an easy way of, you know, programming a computer for a simplexial complex. You just declare what is the rule for simplexial complexes, it understands. Then you give the list of maximal simplices over, okay? So this is, if especially effective by describing a finite simple complex, we need to merely list all maximal simplices. If f is a simplex of a simplexial complex, then the set of all phases of f, okay, just only take the phases of f, including f itself, that itself will be a sub complex, okay? That sub complex also we will denote by f. This is just like when you have a subset of a topological space x, then you write the subspace also by that symbol, same symbol, no? But what is the subspace? It consists of all those open sets inside x, intersection with a. You intersect with a, that is the collection of open subsets of a. You do not keep on writing that, right? So that is like that. So denoted by f itself, okay? The set of all proper phases of f, which means f excluded, that will be also a sub complex. Just the biggest one, the f is not there, but rest of them are there, that is a sub complex again, right? That will be denoted by bf, b corresponding to the boundary. For example, suppose I take a three simplex. A three simplex is a tetrahedral, okay? If I take the boundary, namely the tetrahedral itself is not taken, all subsets are taken. What do I get? All the four triangles I will come. So they form the boundary of the tetrahedral. If you take a simplex, one simplex, what will be its boundary? It will consist of just two singletons, which are the boundary points. So it is like removing the interior? Yeah, now, but we should not speak of that yet because we haven't given any topology. Okay. Yeah, we are removing the whole simplex. The simplex is v1, v2. The subset, it is a sub complex, phases are singleton v0, singleton v1, okay? I am just justifying aim for boundary of f, okay? For this, you can look at the copy of this one, namely standard simplex. What was standard simplex? It was the line joining, delta one, line joining that one. So there we have the topology, right? Though we haven't given topology for all the simplex complex. That is why I have brought this delta n and mod delta n, okay? The motivation is geometric, but definitions are all abstract. Yes, sir. If the dimension of f is n, okay? Then f is a carbon copy of delta n. I am just repeating the same thing that f is isomorphic to delta n. This should already justify to some extent the claim that simplex complex is delta n, delta 1, delta 2, delta 3 and so on. Are the building blocks of simplex complex? Because each simplex there is just a copy of delta n. That n will depend upon dimension, that is all, okay? I think I will stop here. Next time, we will do some more examples. Thank you.